Mapping Mediators and Confounders

Welcome back. In our last session, we constructed the core variables for our experiment, defining the treatment as the four living arrangements and the outcomes as a multi-level satisfaction dashboard. We now have the start and end points of our causal inquiry. Today, we will build the bridges and detours between them.

This lesson focuses on mapping the hypothesized causal structure of the vee triad system. We will identify the mechanisms that explain how living arrangements might affect satisfaction, and the external factors that could distort our measurements. This directly addresses the learning outcome: Map potential mediating pathways (e.g., autonomy, security) and shared confounders (e.g., household stress) onto the system. By the end of this lesson, you will have a foundational Directed Acyclic Graph (DAG) representing our causal assumptions about the triad, a critical step before we can discuss identification and estimation.

1. The Causal Landscape: Mediators and Confounders

Before we construct our full map, let's refresh our understanding of two fundamental concepts—mediators and confounders—and place them within our vee triad context. The short video below provides a clear, concise definition of several key terms in causal modeling.

Confusing Terms in Research - Confounder/Covariate/Moderator/Mediator | Simple Example

This video from the "House of Jia" channel offers a simple, practical overview of terms that are often used interchangeably but have precise causal meanings. We'll use it to quickly frame our discussion.

Please watch the segment defining a confounder (this clip) and the segment defining a mediator (this one). Notice how a confounder is external to the main causal path, while a mediator lies directly upon it.

Mediating Pathways: The "How"

A mediator is a variable on the causal path between an exposure and an outcome; it's the mechanism through which the effect is transmitted. For our experiment, we hypothesize that living arrangements don't just magically alter satisfaction. They change the daily realities of the individuals, which in turn affects their psychological state.

Consider these potential mediating pathways:

Autonomy: A Dispersed Living arrangement (Treatment) might increase an individual's perceived sense of autonomy. This heightened sense of autonomy could then lead to higher role satisfaction (Outcome). The path is: Living Arrangement → Perceived Autonomy → Satisfaction.
Security: A Triadic Cohabitation arrangement might increase feelings of relationship security for all members, which in turn boosts dyadic commitment and overall satisfaction. The path is: Living Arrangement → Felt Security → Satisfaction.
Time & Resource Allocation: The living arrangement directly dictates the division of shared time and resources. A perceived inequity in this allocation could act as a negative mediator, reducing satisfaction. The path: Living Arrangement → Time Allocation → Perceived Equity → Satisfaction.

These are the "frontdoor" paths—the direct causal chains we want to understand.

Shared Confounders: The "Bias"

A confounder is a common cause of both the treatment and the outcome. It creates a "backdoor" path—a non-causal association that can be mistaken for a real effect if not properly handled. Identifying potential confounders is arguably the most critical task in designing an observational study or, in our case, a complex quasi-experiment.

This figure illustrates four common causal structures. Panel A, "Confounding," shows the classic fork structure where a variable (Smoking) is a common cause of the exposure (Exercise) and the outcome (Lung cancer), creating a spurious "back-door path."

In our vee triad scenario, what external factors might influence both the choice of living arrangement and the members' satisfaction levels?

Household Financial Status: A high level of financial stress (Confounder) could both necessitate a cohabiting arrangement to save on costs (Treatment) and independently cause lower life satisfaction for all members (Outcome). If we ignore this, we might wrongly conclude that cohabitation causes dissatisfaction.
External Social Stigma: The degree of disapproval from family, friends, or community (Confounder) could influence the triad to choose a more discreet, dispersed living arrangement (Treatment) while also directly causing psychological distress and lower satisfaction (Outcome).
Pre-existing Relationship Health: The level of commitment and conflict avoidance before the experiment begins (Confounder) will undoubtedly influence which living arrangement is deemed feasible (Treatment) and will also be a strong predictor of future satisfaction (Outcome).

These confounders open backdoor paths that must be blocked through statistical adjustment to isolate the true causal effect of the living arrangement.

2. From Concepts to a Formal Model: Building the DAG

Having identified the key players—treatments, outcomes, mediators, and confounders—we can now assemble them into a formal causal model using a Directed Acyclic Graph. A DAG is more than a sketch; it is a rigorous statement of our causal assumptions.

The following paper provides an excellent framework for applying DAGs to complex social systems. While its focus is on animal social networks, the principles of modeling individual, dyadic, and system-level variables are directly transferable to our human triad.

A causal framework for the drivers of animal social network ...

This paper from PLOS Computational Biology demonstrates a formal workflow for moving from causal hypotheses to a testable statistical model in a social context. We will adapt its logic to structure our own model.

First, briefly review the "Introduction to directed acyclic graphs" subsection under "Simulation study 1" to refamiliarize yourself with the formal components of a DAG, such as paths and influence. Next, focus on the "Empirical application" section, specifically the subsection on the Directed acyclic graph for the macaque grooming example. As you read, note how the authors build a model for a complex social behavior: Identify the main causal relationship of interest (the "frontdoor path"): Kinship → Affiliation. This is analogous to our Living Arrangement → Satisfaction path. Identify the confounder: Dominance Rank. Notice how it's a common cause of both the exposure and the outcome. Recognize the backdoor path this confounder creates: the path from Kinship to Grooming Rate that does not represent the direct effect. This is precisely the kind of spurious path we must identify in our own system.

The video below offers a clear, animated explanation of backdoor paths and how they are "blocked" by conditioning on a set of variables. This is the fundamental reason we are so concerned with identifying confounders.

Causal Effects via DAGs | How to Handle Unobserved Confounders

This video by Shaw Talebi provides an excellent visual walkthrough of the graphical criteria for causal identification.

Watch the section from this timestamp that defines a backdoor path and the concept of blocking it with a chain or a fork. This visualizes the problem that confounders create.

Our Triad's Initial DAG

Let's synthesize these ideas into a first-pass DAG for our vee triad. Let P be the Pivot, W1 be Wing 1, and W2 be Wing 2.

Nodes:

T: Treatment (Living Arrangement, categorical variable {A,B,C,D})
Y_P, Y_W1, Y_W2: Individual satisfaction outcomes for each person.
C_Fin: A shared confounder (e.g., Household Financial Stress).
M_Aut_P: A mediating variable (e.g., Pivot's Perceived Autonomy).

A Simplified Causal Structure:

Main Effect Path: T → M_Aut_P → Y_P (Living arrangement affects the Pivot's autonomy, which in turn affects their satisfaction).
Confounding Path: We have a fork. C_Fin → T (Finances affect the choice of living arrangement) and C_Fin → Y_P, C_Fin → Y_W1, C_Fin → Y_W2 (Finances also directly affect everyone's satisfaction). This creates a backdoor path from T to each Y, e.g., T ← C_Fin → Y_P.
Interference Paths: The satisfaction of one member can influence another's. We can represent this with arrows like Y_P → Y_W1 and Y_W1 → Y_P (acknowledging this would need to be time-indexed in a full model to remain acyclic, e.g., Y_P(t) -> Y_W1(t+1)).

This simple DAG already makes our assumptions explicit and highlights the necessity of measuring and adjusting for C_Fin to isolate the true effect of T on Y.

3. A Deeper Dive: Confounding in Networked Systems

In an interconnected system like our triad, the concept of a "confounder" can be more nuanced. Not all unmeasured factors affect all members equally or in the same way. Your background in SUTVA violations and interference makes this an important point to consider. A recent paper in causal representation learning formalizes this by categorizing latent confounders in a network.

[PDF] Causal Effect Estimation under Networked Interference without ...

This paper addresses the problem of unmeasured confounding in networked settings by proposing a typology of latent confounders. This gives us a more sophisticated language for mapping the sources of bias in our triad.

Please read the Introduction and pay close attention to Figure 2 and the definitions of the three categories of latent confounders. Focus on understanding the distinction between: The three types: ui (unit-specific), un (neighbor-specific), and uc (common/shared). Assumption 5: The formal statement that decomposes the total set of confounders into these distinct types.

Let's apply this ui, un, uc framework to the Pivot (P) as our focal unit:

uc (Common Confounder): This is the shared confounder we've already discussed. Household Financial Stress is a perfect example. It's a single variable that is a common cause of the treatment T and the outcomes of all three members (Y_P, Y_W1, Y_W2). In the paper's notation, this uc affects the unit (P) and its neighbors (W1 and W2).
ui (Unit-specific Confounder): This is a confounder that affects only the focal unit's treatment assignment and outcome. Imagine the Pivot develops a chronic health issue. This health issue (ui) might make them prefer a living arrangement where they have more support (T), and it would also directly affect their own satisfaction (Y_P). However, it wouldn't be a direct cause of W1's or W2's satisfaction (though it might affect them indirectly through P's behavior, which is interference, not confounding).
un (Neighbor-specific Confounder): This is the most subtle type. It's a factor related to a neighbor that confounds the relationship between the system-wide treatment and the focal unit's outcome. Let's say Wing 1's job becomes unstable. This instability (un from P's perspective) might push the triad to avoid the P-W1 Dyadic Cohabitation arrangement (T). At the same time, this job instability could independently create stress that spills over and affects the Pivot's satisfaction (Y_P). This un is a common cause of T and Y_P, but it originates with a neighbor.

This refined view of confounding is not just theoretical. It forces us to think more precisely about the sources of bias and what, specifically, we would need to measure to control for them. Our full DAG must eventually account for all three types.

Conclusion

In this lesson, we have laid the intellectual groundwork for our causal analysis by constructing a preliminary map of the triadic system. We have moved beyond simple Treatment → Outcome thinking to incorporate the complexities of social dynamics.

The key takeaways are:

Causal maps require both mediators and confounders. Mediators explain the mechanisms of an effect, while confounders represent sources of spurious correlation that must be controlled.
DAGs are the formal language for expressing causal assumptions. They make our hypotheses transparent and provide a non-negotiable guide for the analysis phase.
Confounding in a network is complex. Distinguishing between unit-specific (ui), neighbor-specific (un), and common (uc) confounders allows for a more precise model of bias in an interconnected system.

We have now identified the essential variables we need to consider: the treatments, the multi-level outcomes, the mediating pathways, and the various types of confounders. The natural question that follows is: how do we actually measure these things?

In our next lesson, we will tackle this question directly as we move to the next learning outcome: Design a multi-dimensional satisfaction instrument tailored to the vee roles, mitigating potential floor and ceiling effects. We will discuss the practical challenges of creating valid and reliable measurement tools for these complex, latent psychological constructs.

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